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Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves

Ildefonso Castro, Ildefonso Castro-Infantes, Jesús Castro-Infantes

Abstract

We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.

Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves

Abstract

We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.
Paper Structure (9 sections, 9 theorems, 69 equations, 4 figures)

This paper contains 9 sections, 9 theorems, 69 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Helicoidal and rotational surfaces in the 3-sphere
  3. The spherical angular momentum of a spherical curve
  4. Spherical small circles
  5. Spherical catenaries
  6. Prescribing curvature for a helicoidal surface in $\mathbb{S}^3$
  7. Rotational and helicoidal minimal surfaces in $\mathbb{S}^3$
  8. Rotational minimal surfaces in $\mathbb{S}^3$
  9. Helicoidal minimal surfaces in $\mathbb{S}^3$

Key Result

Theorem 3.1

Any spherical curve $\xi=(x,y,z):I\subseteq \mathbb{R} \rightarrow \mathbb{S}^2$, with $z$ non-constant, is uniquely determined by its spherical angular momentum $\mathcal{K}$ as a function of its coordinate $z$, that is, by $\mathcal{K}= \mathcal{K}(z)$. The uniqueness is modulo rotations around th

Figures (4)

  • Figure 1: Small circles: $\mathcal{K} (z)= k_0 \, z + c, \ k_0 >0$; $0\leq |c|<1$ (left), $c=\pm 1$ (center), $1<|c|<\sqrt{1+k_0^2}$ (right).
  • Figure 2: Two sights of the spherical catenaries $\mathcal{C}_{\beta_{2/3}}$ (left), $\mathcal{C}_{\beta_{3/5}}$ (center) and $\mathcal{C}_{\beta_{5/8}}$ (right).
  • Figure 3: Open sights (with $t\in (0,3\pi/2)$) of the Otsuki-Brendle-Kusner spherical catenoids $\mathrm{Cat}_{\beta_{2/3}}$ and $\mathrm{Cat}_{\beta_{3/5}}$.
  • Figure 4: Representation of the associated immersions to a spherical catenoid.

Theorems & Definitions (30)

  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Example 3.3: $\kappa\!\equiv\! 0$
  • Corollary 3.4
  • ...and 20 more